Whisk together flour, baking powder, baking soda and spices. Beat butter, brown sugar, and vanilla with an electric mixer at high speed until pale and fluffy, 2 to 3 minutes. Add eggs 1 at a time, beating well after each addition, then beat in applesauce. At low speed, mix in flour mixture until just combined.

Spread batter evenly in pan and bake until golden-brown and a wooden pick inserted into center comes out clean, 40 to 45 minutes. Cool in pan 15 minutes. Run a knife around edge of cake to loosen, then invert onto a plate. Reinvert cake onto a rack to cool completely. Dust with powdered sugar before serving.